Pricing Synthetic CDOs based on Exponential

Approximations to the Payoff Function∗

Ian Iscoe† Ken Jackson‡ Alex Kreinin§ Xiaofang Ma¶

April 4, 2011

Abstract

Correlation-dependent derivatives, such as Asset-Backed Securities (ABS) and Col-

lateralized Debt Obligations (CDO), are common tools for offsetting credit risk. Factor

models in the conditional independence framework are widely used in practice to capture

the correlated default events of the underlying obligors. An essential part of these models

is the accurate and efficient evaluation of the expected loss of the specified tranche, con-

ditional on a given value of a systematic factor (or values of a set of systematic factors).

Unlike other papers that focus on how to evaluate the loss distribution of the underlying

pool, in this paper we focus on the tranche loss function itself. It is approximated by a

sum of exponentials so that the conditional expectation can be evaluated in closed form

without having to evaluate the pool loss distribution. As an example, we apply this

approach to synthetic CDO pricing. Numerical results show that it is efficient.

∗This research was supported in part by the Natural Sciences and Engineering Research Council (NSERC)

of Canada.†Algorithmics Inc., 185 Spadina Avenue, Toronto, ON, M5T 2C6, Canada; Ian.Iscoe@algorithmics.com‡Department of Computer Science, University of Toronto, 10 King’s College Rd, Toronto, ON, M5S 3G4,

Canada; krj@cs.utoronto.ca§Algorithmics Inc., 185 Spadina Avenue, Toronto, ON, M5T 2C6, Canada; Alex.Kreinin@algorithmics.com¶Risk Services, CIBC, 11th Floor, 161 Bay Street, Toronto, ON, M5X 2S8, Canada; Xiaofang.Ma@cibc.com

1

1 Introduction

Correlation-dependent derivatives, such as Asset-Backed Securities (ABS) and Collateralized

Debt Obligations (CDO), are common tools for offsetting credit risk.1 Factor models in the

conditional independence framework are widely used in practice for pricing because they are

tractable, analytic or semi-analytic formulas being available. An essential part of these models

is the accurate and efficient evaluation of the expected loss of the specified tranche, conditional

on a given value of a systematic factor (or correspondingly values of a set of systematic factors).

To be specific, the problem is how to evaluate the conditional expectation of the piecewise

linear payoff function of the loss Z

f(Z) = min(

u − ℓ, (Z − ℓ)+)

, (1)

where x+ = max(x, 0), Z =∑K

k=1 Zk, Zk are conditionally mutually independent, but not nec-

essarily identically distributed, nonnegative random variables in a conditional independence

framework (see Section 2 for explanations), and ℓ and u are the attachment and the detach-

ment points of the tranche, respectively, satisfying u > ℓ ≥ 0. Generally Zk, for obligor k, is

the product of the two components: a random variable directly related to its credit rating and

a loss-given-default or mark-to-market related value. The payoff function, f , is also known as

the stop-loss function in actuarial science [4], [23].

Note that the expectation of a function of a random variable depends on two things: the

distribution of the underlying random variable and the function itself. A standard approach

to compute the expectation of a function of a random variable is to compute firstly the

distribution of the underlying random variable, Z in our case, and then to compute the

expectation of the given function, possibly using its special properties. Almost all research in

finance [2], [9], [10], [14], [22], [25] and in actuarial science [7], [28], to name a few, has focused

on the first part due to the piecewise linearity of the payoff function.

In this paper, we propose a new approach to solving the problem. We approximate the

1For background on, and a description of, these instruments, see for example [5]. We restrict attention to

synthetic CDOs in this paper.

2

non-smooth function f by a sum of exponentials over [0,∞). Based on this approximation, the

conditional expectation can be computed from a series of simple expectations. Consequently,

we do not need to compute the distribution of Z. Due to the central role played by the

exponential approximation to the payoff function in our method, we refer to it as the EAP

method.

The remainder of this paper is organized as follows. The details of our approach out-

lined above are described in Section 2. As an example, this method is applied to synthetic

CDO pricing in Section 3. The paper ends with some conclusions in Section 4, in which we

summarize the advantages of our method over others, and indicate its scope of applicability.

2 Conditional expectation based on an exponential ap-

proximation

In the conditional independence framework, a central problem is how to evaluate the expec-

tation

E[f(Z)] =

∫

M

EM [f (Z)] dΦ(M),

where Φ(M) is the distribution of an auxiliary factor M (which can be a scalar or a vector),

EM [f (Z)] ≡ E [f (Z) |M = M ]

and

Z =K∑

k=1

Zk, (2)

where Zk ≥ 0 are mutually independent random variables, conditional on M . It is obvious

that Z is nonnegative. We denote by ΨM the distribution of Z conditional on M = M , so

that

EM [f (Z)] =

∫

z

f (z) dΨM(z). (3)

Due to the piecewise linearity of the function f defined by (1), it is clear that once the

distribution ΨM is obtained, the conditional expectation∫

zf (z) dΨM(z) can be readily com-

3

puted. Most research has focused on how to evaluate the conditional distribution of Z given

the conditional distributions of Zk. A fundamental result about a sum of independent random

variables states that Z’s conditional distribution can be computed as the convolution of Zk’s

conditional distributions. Numerically, this idea is realized through forward and inverse fast

Fourier transformations (FFT). A disadvantage of this approach is that it may be very slow

when there are many obligors due to the number of convolutions to be calculated. For pools

with special structures, it might be much slower than methods that are specially designed

for those pools, such as recursive methods proposed by De Pril [7] and Panjer [28] and their

extensions discussed in [4] and [23], and the one proposed by Jackson, Kreinin, and Ma [22]

for portfolios where the Zk sit on a properly chosen common lattice. To avoid computing too

many convolutions, the target conditional distribution can be approximated by parametric

distributions matching the first few conditional moments of the true conditional loss distri-

bution. For a large pool, a normal approximation is a natural choice as a consequence of the

central limit theorem and due to its simplicity, although it may not capture some important

properties, such as skewness and fat tails.

To capture these important properties for medium to large portfolios, compound approx-

imations, such as the compound Poisson [20], improved compound Poisson [13], compound

binomial and compound Bernoulli [29] distributions have been used. They have proved to be

very successful, since they match not only the first few conditional moments, but, most im-

portantly, they match the tails much better than either normal or normal power distributions

do. A key step in a method based on these compound approximations is the computation of

convolutions of the conditionally independent distributions of the Zk, by FFTs. As a result,

the computational complexity of such an algorithm is superlinear in K, the number of terms

in the sum (2).

As an alternative, in this paper, we propose an algorithm for which the computational

complexity is linear in K. We focus on the stop-loss function f , instead of the conditional

distribution ΨM of Z. To emphasis the role of the attachment and the detachment points ℓ

and u, we denote f(x) by f(x; ℓ, u) and introduce an auxiliary function h(x) defined on [0,∞):

4

h(x) = 1 − x if x ≤ 1, 0 otherwise. Then we have

f(x) = f(x; ℓ, u) = u[

1 − h(x

u

)]

− ℓ[

1 − h(x

ℓ

)]

. (4)

In particular, if ℓ = 0, we have

f(x; 0, u) = min(

u, x+)

= min (u, x) = u[

1 − h(x

u

)]

.

Note that h(x) is independent of the constants ℓ and u. Therefore, if it can be approxi-

mated by a sum of exponentials over [0,∞), it is clear that f(x; ℓ, u) can be approximated by

a sum of exponentials. Let

h(x) ≈

N∑

n=1

ωn exp(γnx), (5)

where ωn and γn are complex numbers. Then from (4) we can see that f(x; ℓ, u) can be

approximated by a sum of exponentials:

f(x; ℓ, u) ≈ u

[

1 −N∑

n=1

ωn exp(

γn

x

u

)

]

− ℓ

[

1 −N∑

n=1

ωn exp(

γn

x

ℓ

)

]

(6)

≈ (u − ℓ) − u

N∑

n=1

ωn exp(γn

ux)

+ ℓ

N∑

n=1

ωn exp(γn

ℓx)

. (7)

Based on this expression the conditional expectation EM [f (Z)] defined in (3) can be

computed as follows:

5

EM [f (Z)]

=

∫

z

f (z) dΨM(z)

≈

∫

z

[

(u − ℓ) − u

N∑

n=1

ωn exp(γn

uz)

+ ℓ

N∑

n=1

ωn exp(γn

ℓz)

]

dΨM(z)

= (u − ℓ) − uN∑

n=1

ωn

∫

z

exp(γn

uz)

dΨM(z)

+ℓ

N∑

n=1

ωn

∫

z

exp(γn

ℓz)

dΨM(z)

= (u − ℓ)

−uN∑

n=1

ωn

∫

z1,...,zK

K∏

k=1

exp(γn

uzk

)

dΨM,1(z1) · · ·dΨM,K(zK)

+ℓ

N∑

n=1

ωn

∫

z1,...,zK

K∏

k=1

exp(γn

ℓzk

)

dΨM,1(z1) · · ·dΨM,K(zK)

= (u − ℓ) − uN∑

n=1

ωn

K∏

k=1

EM

[

exp(γn

uZk

)]

+ℓ

N∑

n=1

ωn

K∏

k=1

EM

[

exp(γn

ℓZk

)]

, (8)

where ΨM,k is the conditional distribution of Zk, EM [exp (cZk)] is the conditional expectation

of exp (cZk), for c = γn

ℓor γn

u, respectively. The last equality holds by noting that Zk, thus

cZk, are mutually independent conditional on a given value of M . In this way we can see that,

to compute the conditional expectation EM [f (Z)], we only need to compute the conditional

expectations EM [exp (cZk)] of individual obligors.

The approximation in (5) is understood in the uniform sense and the uniformity is inherited

in (7). Denoting the function on the right-hand side of (7) by f , then the right-hand side of

(8) is EM [f(Z)]. If supz≥0 |h(z)− h(z)| ≤ ε/(u + ℓ) then, by (6), supz≥0 |f(z)− f(z)| ≤ ε, and

hence

|EM [f(z)] − EM [f(z)]| ≤ ε (9)

and similarly for the unconditional expectations.

6

Since h(z) is independent of the constants ℓ and u, for a given approximation accuracy

the coefficients ωn and γn for (5) need to be computed only once and the number of terms

required can be determined a priori. As shown in a separate paper by the authors [17], the

maximum absolute error in the approximation (5) is roughly equal to 0.16/N :

Table 1: The maximum absolute error in the approximation (5) for several values of N

N 25 50 100 200 400

Max absolute error 6.4×10−3 3.2×10−3 1.6×10−3 8×10−4 4×10−4

This numerical result leads to the following rule of thumb for a practical choice of N . To

achieve a desired accuracy ε in the calculation of an expected tranche loss, with attachment

and detachment points ℓ and u, respectively, one can choose N > 0.16(u + ℓ)/ε. As such,

from Table 1, supz≥0 |h(z) − h(z)| ≤ ε/(u + ℓ) and then (9) holds. If ℓ and u are expressed

as percentages of the total pool notional, then u + ℓ ≤ 2 and we can simply take N > 0.32/ε

to obtain a maximum relative error of size ε in the calculation of any expected tranche loss,

relative to the total pool notional.

As an example, the parameters γn and ωn for a 25-term approximation are shown in

Table 2. It is proved in [17] that, if γn is real, then ωn is also real, and if γi and γj are a

complex conjugate pair, then the corresponding ωi and ωj are also a complex conjugate pair,

and vice versa. The data in the table exhibit this property. Exploiting this property, we can

simplify the summations in (8) by noting that the sum of the complex conjugated i-th and j-th

terms equals twice the real part of either one of these two terms. The table also shows that

the real part of each γn is strictly negative. This property guarantees that the exponential

approximation of (5) converges to zero as x → ∞, and thus guarantees the existence of the

conditional expectation EM [exp (cZk)]. For a more complete discussion of the exponential

approximation (5), see [17]. In particular, it is proved therein, that each γn has a strictly

negative real part for any N .

7

Table 2: The real and imaginary parts of ωn and γn (weights and exponents) for the 25-term

exponential approximation

n ℜ(ωn) ℑ(ωn) ℜ(γn) ℑ(γn)

1 1.68E-4 -3.16E-5 -5.68E-2 1.45E+2

2 1.68E-4 3.16E-5 -5.68E-2 -1.45E+2

3 2.04E-4 -5.98E-5 -1.72E-1 1.32E+2

4 2.04E-4 5.98E-5 -1.72E-1 -1.32E+2

5 2.69E-4 -1.02E-4 -3.51E-1 1.20E+2

6 2.69E-4 1.02E-4 -3.51E-1 -1.20E+2

7 3.87E-4 -1.70E-4 -5.98E-1 1.07E+2

8 3.87E-4 1.70E-4 -5.98E-1 -1.07E+2

9 6.02E-4 -2.94E-4 -9.25E-1 9.49E+1

10 6.02E-4 2.94E-4 -9.25E-1 -9.49E+1

11 1.01E-3 -5.39E-4 -1.35E+0 8.24E+1

12 1.01E-3 5.39E-4 -1.35E+0 -8.24E+1

13 1.87E-3 -1.09E-3 -1.89E+0 6.99E+1

14 1.87E-3 1.09E-3 -1.89E+0 -6.99E+1

15 3.86E-3 -2.53E-3 -2.58E+0 5.74E+1

16 3.86E-3 2.53E-3 -2.58E+0 -5.74E+1

17 9.17E-3 -7.44E-3 -3.50E+0 4.49E+1

18 9.17E-3 7.44E-3 -3.50E+0 -4.49E+1

19 2.45E-2 -3.19E-2 -4.73E+0 3.24E+1

20 2.45E-2 3.19E-2 -4.73E+0 -3.24E+1

21 7.57E-3 -2.10E-1 -6.44E+0 2.04E+1

22 7.57E-3 2.10E-1 -6.44E+0 -2.04E+1

23 3.81E+0 0.00E+0 -9.65E+0 0.00E+0

24 -1.46E+0 1.03E-1 -8.54E+0 9.38E+0

25 -1.46E+0 -1.03E-1 -8.54E+0 -9.38E+0

8

3 Pricing synthetic CDOs

We illustrate the EAP method by applying it to synthetic CDO pricing. The underlying

collateral of a synthetic CDO is a set of credit default swaps (CDSs).

3.1 Overview of models and pricing methods

Factor models, such as the reduced-form model proposed by Laurent and Gregory [25] or the

one proposed by Duffie and Garleanu [10] and extended successively by Mortensen [27] and

Eckner [11], and the structural model proposed by Vasicek [31] and Li [26], are widely used

in practice to obtain analytic or semi-analytic formulas to price synthetic CDOs efficiently.

For a comparative analysis of different copula models, see the paper by Burtschell, Gregory

and Laurent [6]. A generalization of the reduced-form model which, along with the structural

model, can capture credit-state migration, is the affine Markov model proposed by Hurd and

Kuznetsov [16] and used by them in pricing CDOs in [15].

Some exact and approximate methods for loss-distribution evaluation have been studied in

[22]. There are also Fourier transform related methods. In particular, the saddle point approx-

imation method2 was applied by Antonov, Mechkov, and Misirpashaev [3] and Yang, Hurd

and Zhang [32], for pricing CDOs. Here we apply the exponential-approximation method, as

described in the previous section, to synthetic CDO pricing.

3.2 The pricing equation and the Gaussian copula model

To illustrate our method, we use a simple one-factor Gaussian copula model. Examples of

other models, to which the EAP method is applicable, are given in the Conclusion, Section 4.

It is assumed that the interest rates are deterministic and the recovery rate corresponding to

2In principle, this method is applicable to large pools, as the pool size plays the role of the large parameter

(albeit not in the classical multiplicative manner) in the inverse Fourier integrand’s exponent to which the

saddle point method is applied.

9

each underlying name is constant. Let 0 < t1 < t2 < · · · < tn = T be the set of premium

dates, with T denoting the maturity of the CDO, and d1, d2, . . . , dn be the set of corresponding

discount factors. Suppose there are K names in the pool with recovery-adjusted notional

values LGD1, LGD2, . . . , LGDK in properly chosen units. Let L Pi be the pool’s cumulative

losses up to time ti and ℓ be the attachment point of a specified tranche of thickness S. An

attachment point of a tranche is a threshold that determines whether some losses of the pool

shall be absorbed by this tranche, i.e., if the realized losses of the pool are less than ℓ, then

the tranche will not suffer any loss, otherwise it will absorb an amount up to S. Accordingly,

the detachment point of the tranche is u = S + ℓ. Thus the loss absorbed by the specified

tranche up to time ti is Li = min(

S, (L Pi − ℓ)+

)

. If we further assume the fair spread s for

the tranche is a constant, then it can be calculated from the equation (see e.g., [8], [20])

s =E [∑n

i=1(Li − Li−1)di]

E [∑n

i=1(S − Li)(ti − ti−1)di]=

∑n

i=1 E [(Li − Li−1)di]∑n

i=1 E [(S − Li)(ti − ti−1)di], (10)

with t0 = 0 and E [L0] = 0.

Assuming the discount factors di are independent of the L Pj , it follows from (10) that the

problem of computing the fair spread s reduces to evaluating the expected cumulative losses

E [Li], i = 1, 2, . . . , n. In order to compute this expectation, we have to specify the default

processes for each name and the correlation structure of the default events. One-factor models

were first introduced by Vasicek [31] to estimate the loan loss distribution and then generalized

by Li [26], Gordy and Jones [12], Hull and White [14], Iscoe, Kreinin and Rosen [21], Laurent

and Gregory [25], and Schonbucher [30], to name a few.

Let τk be the default time of name k. Assume the risk-neutral default probabilities πk(t) =

P (τk < t), k = 1, 2, . . . , K, that describe the default-time distributions of all K names are

available, where τk and t take discrete values from {t1, t2, . . . , tn}. The dependence structure

of the default times are determined in terms of their creditworthiness indices Yk, which are

defined by

Yk = βkX + σkεk, k = 1, 2, . . . , K, (11)

where X is the systematic risk factor, εk are idiosyncratic factors that are mutually indepen-

dent and are also independent of X; βk and σk are constants satisfying the relation β2k +σ2

k = 1.

10

These risk-neutral default probabilities and the creditworthiness indices are related by the

threshold model

πk(t) = P (τk < t) = P (Yk < Hk(t)) , (12)

where Hk(t) is the default barrier of the k-th name at time t.

Thus the correlation structure of default events is captured by the systematic risk factor X.

Conditional on a given value x of X, all default events are independent. If we further assume,

as we do, that X and εk follow the standard normal distribution, then Yk is a standard normal

random variable and from (12) we have Hk(t) = Φ−1 (πk(t)). Furthermore, the correlation

between two different indices Yi and Yj is βiβj .

The conditional, risk-neutral default-time distribution is defined by

πk(t; x) = P (Yk < Hk(t)|X = x) . (13)

Thus from (11) and (13) we have

πk(t; x) = Φ

(

Hk(t) − βkx

σk

)

. (14)

The conditional and unconditional risk-neutral default-time probabilities at the premium date

ti are denoted by πk(i; x) and πk(i), respectively.

In this conditional independence framework, the expected cumulative tranche losses E [Li]

can be computed as

E [Li] =

∫ ∞

−∞

Ex [Li] dΦ(x), (15)

where Ex [Li] = Ex

[

min(

S, (L Pi − ℓ)+

)]

is the expectation of Li conditional on the value of

X being x, where L Pi =

∑K

k=1 LGDk1{Yk<Hk(ti)}, and the indicators 1{Yk<Hk(ti)} are mutu-

ally independent conditional on X. Generally, the integration in (15) needs to be evaluated

numerically using an appropriate quadrature rule:

E [Li] ≈M∑

m=1

wmExm

[

min(

S, (L Pi − ℓ)+

)]

. (16)

In the notation of Section 2, Z = L Pi , Zk = LGDk1{Yk<Hk(ti)}, and M = X.

11

3.3 CDO pricing based on the exponential approximation

Notice from formula (16) that the fundamental problem in CDO pricing is how to evaluate

the conditional expected loss Exm

[

min(

S, (L Pi − ℓ)+

)]

with a given value xm of X. Since

min(

S, (L Pi − ℓ)+

)

= f(L Pi ; ℓ, ℓ + S), (17)

from (7) we see that

min(

S, (L Pi − ℓ)+

)

≈(ℓ + S)

[

1 −N∑

n=1

ωn exp

(

γn

L Pi

ℓ + S

)

]

− ℓ

[

1 −N∑

n=1

ωn exp

(

γn

L Pi

ℓ

)

]

=S − (ℓ + S)

N∑

n=1

ωn exp

(

γn

ℓ + SL

Pi

)

+ ℓ

N∑

n=1

ωn exp(γn

ℓL

Pi

)

.

As a special case of (8) we have

Exm

[

min(

S, (L Pi − ℓ)+

)]

≈S − (ℓ + S)

N∑

n=1

ωnExm

[

exp

(

γn

ℓ + S

K∑

k=1

LGDk1{Yk<Hk(ti)}

)]

+ ℓN∑

n=1

ωnExm

[

exp

(

γn

ℓ

K∑

k=1

LGDk1{Yk<Hk(ti)}

)]

=S − (ℓ + S)

N∑

n=1

ωn

K∏

k=1

Exm

[

exp

(

γn

ℓ + SLGDk1{Yk<Hk(ti)}

)]

+ ℓN∑

n=1

ωn

K∏

k=1

Exm

[

exp(γn

ℓLGDk1{Yk<Hk(ti)}

)]

, (18)

where

Exm

[

exp

(

γn

ℓ + SLGDk1{Yk<Hk(ti)}

)]

= πk(i; xm) exp

(

γn

ℓ + SLGDk

)

+ (1 − πk(i; xm)) ,

Exm

[

exp(γn

ℓLGDk1{Yk<Hk(ti)}

)]

= πk(i; xm) exp(γn

ℓLGDk

)

+ (1 − πk(i; xm)) ,

since 1{Yk<Hk(ti)} = 1 with probability πk(i; xm) and 0 with probability 1 − πk(i; xm), and

LGDk is a constant.

3.4 Numerical results

In this section we present numerical results comparing the accuracy and the computational

time for EAP and the exact method, which we denote by JKM, proposed in [22]. The results

12

presented below are based on a sample of 15 = 3×5 pools: three pool sizes (100, 200, 400) and,

for each size, five different types in terms of the notional values of the underlying reference

entities. The possible notional values for each pool are summarized in Table 3. The number

of reference entities is the same in each of the homogeneous subgroups in a given pool. For

example, for a 200-name pool of type 3, there are four homogeneous subgroups each with

50 reference entities of notional values 50, 100, 150, and 200, respectively. Thus, the higher

the pool type, the more heterogeneous is the pool, with pool type 1 indicating a completely

homogeneous pool in terms of notional values. In the figures and tables that follow, the label

K-m denotes a pool of size K and pool type m.

Table 3: Selection of notional values of K-name pools

Pool Type 1 2 3 4 5

Notional values 100 50, 100 50, 100, 150, 200 20, 50, 100, 150, 200 10, 20, . . . , K

For each name, the common risk-neutral cumulative default probabilities are given in

Table 4.

Table 4: Risk-neutral cumulative default probabilities

Time 1 yr. 2 yrs. 3 yrs. 4 yrs. 5 yrs.

Prob. 0.0072 0.0185 0.0328 0.0495 0.0680

The recovery rate is assumed to be 40% for all names. Thus the LGD of name k is

0.6Nk. The maturity of a CDO deal is five years (i.e., T = 5) and the premium dates are

ti = i, i = 1, . . . , 5 years from today (t0 = 0). The continuously compounded interest rates

are r1 = 4.6%, r2 = 5%, r3 = 5.6%, r4 = 5.8% and r5 = 6%. Thus the corresponding discount

factors, defined by di = exp(−tiri), are 0.9550, 0.9048, 0.8454, 0.7929 and 0.7408, respectively.

All CDO pools have five tranches that are determined by the attachment points (ℓ’s) of the

tranches. For this experiment, the five attachment points are: 0, 3%, 7%, 10% and 15%, with

the last detachment point being 30%. The constants βk are all 0.5. (In practice, the βk’s are

13

known as sensitivities and are calibrated to market data. Here they are simply taken as input

to the model, for the purpose of numerical experimentation.)

All methods for this experiment were coded in Matlab and the programs were run on a

Pentium D 925 PC. The results are presented in Tables 5, 6 and 7.

The accuracy comparison results are presented in Table 5. The four numbers in each

pair of brackets in the main part of the table are the spreads in basis points for the first

four tranches of the corresponding pool. For example, (2248.16, 635.22, 268.22, 118.34) are the

spreads evaluated by the JKM method for the first four tranches of the 200-name homogeneous

pool with type 1 reference entity composition. The entries under “25-term”, “100-term”, and

“400-term” are the spreads evaluated using the exponential-approximation method with 25,

100 and 400 terms, respectively. From the table we can see that, as the number of terms

increases, the accuracy of the spreads improves.

As a side-experiment, a comparison was made between the spreads for the third tranche,

[7%, 10%], and its thickening, [7%, 10.1%], to check for monotonicity. In all cases, the spreads

decreased (as they should) and by the amounts shown in Table 6. For the EAP method,

25 terms were used, as that represents the least accurate of the approximations used in the

experiments and therefore provides the most stringent test.

The ratio of CPU times used by the JKM method and the exponential-approximation

method using different numbers of terms, for the 15 test pools are presented in Table 7. The

numbers in the table represent the ratio, EAP/JKM, of average timings over 50 runs, for the

calcuation of the spread for the fourth tranche, [10%, 15%]. Note that, for the exponential-

approximation method, its CPU time is independent of the pool structure: its computational

cost depends only on the number of names and the number of terms in the exponential

approximation. This feature is not shared by most of the other commonly used methods for

pricing CDOs. From the table we see that, as the pool becomes more heterogeneous, the

ratio of (EAP time)/(JKM time) decreases; as the pool becomes larger, the ratio of (EAP

time)/(JKM time) decreases. It is interesting to note from (7) that, for a given pool, to

evaluate any single tranche using the exponential-approximation method takes about as much

14

Table 5: Accuracy comparison between the exact JKM method and the EAP method using 25, 100 and 400 terms

Pool ID Exact 25-term 100-term 400-term

100-1 (2167.69, 642.44, 276.38, 123.50) (2165.42, 643.11, 276.90, 123.25) (2167.77, 642.49, 276.30, 123.47) (2167.73, 642.44, 276.38, 123.50)

100-2 (2142.13, 647.07, 278.40, 124.34) (2141.24, 647.51, 278.12, 124.6) (2142.41, 647.03, 278.29, 124.45) (2142.20, 647.06, 278.38, 124.35)

100-3 (2128.39, 648.42, 279.39, 125.38) (2128.97, 648.18, 279.51, 125.56) (2128.52, 648.44, 279.44, 125.38) (2128.44, 648.42, 279.39, 125.39)

100-4 (2097.58, 651.38, 282.49, 127.35) (2097.14, 651.60, 282.59, 127.57) (2097.61, 651.45, 282.49, 127.33) (2097.6, 651.39, 282.50, 127.35)

100-5 (3069.39, 688.44, 167.46, 35.80) (3069.46, 688.63, 167.44, 35.88) (3069.51, 688.44, 167.47, 35.80) (3069.43, 688.44, 167.46, 35.81)

200-1 (2248.16, 635.22, 268.22, 118.34) (2246.75, 635.79, 268.36, 118.55) (2248.17, 635.29, 268.22, 118.28) (2248.18, 635.22, 268.22, 118.34)

200-2 (2237.60, 636.69, 269.06, 118.85) (2236.93, 636.98, 269.28, 119.09) (2237.69, 636.69, 268.99, 118.91) (2237.62, 636.69, 269.06, 118.85)

200-3 (2229.45, 637.58, 269.84, 119.32) (2229.11, 637.87, 270.00, 119.53) (2229.45, 637.63, 269.85, 119.32) (2229.46, 637.58, 269.84, 119.32)

200-4 (2212.52, 639.43, 271.42, 120.30) (2212.44, 639.65, 271.62, 120.48) (2212.59, 639.48, 271.39, 120.30) (2212.53, 639.44, 271.42, 120.3)

200-5 (3350.42, 662.54, 136.15, 24.47) (3350.43, 662.82, 136.14, 24.51) (3350.50, 662.54, 136.14, 24.47) (3350.43, 662.54, 136.15, 24.47)

400-1 (2291.12, 30.91, 264.05, 115.78) (2290.68, 631.3, 264.34, 116.07) (2291.17, 630.99, 263.98, 115.77) (2291.12, 630.92, 264.05, 115.78)

400-1 (2285.92, 31.56, 264.50, 116.05) (2285.76, 631.84, 264.79, 116.35) (2285.95, 631.57, 264.51, 116.04) (2285.93, 631.56, 264.50, 116.06)

400-3 (2281.84, 632.00, 264.88, 116.29) (2281.82, 632.26, 265.15, 116.57) (2281.82, 632.05, 264.88, 116.26) (2281.85, 632.01, 264.89, 116.29)

400-4 (2273.15, 632.96, 265.69, 116.78) (2273.19, 633.20, 265.95, 117.05) (2273.22, 632.96, 265.69, 116.76) (2273.16, 632.96, 265.69, 116.78)

400-5 (3427.70, 649.51, 130.86, 23.59) (3427.86, 649.78, 130.87, 23.65) (3427.79, 649.50, 130.85, 23.58) (3427.71, 649.51, 130.86, 23.59)

15

Table 6: Tranche spread-differences between tranches [7, 10] and [7, 10.1], to check monotonic-

ity of spreads; 25 terms used in EAP method

Pool ID JKM EAP-25

100-1 2.46 2.69

100-2 2.88 2.66

100-3 2.62 2.67

100-4 2.69 2.69

100-5 2.73 2.73

200-1 2.62 2.62

200-2 2.69 2.62

200-3 2.62 2.63

200-4 2.63 2.64

200-5 2.42 2.42

400-1 2.60 2.60

400-2 2.60 2.61

400-3 2.60 2.61

400-4 2.61 2.61

400-5 2.33 2.33

time as to evaluate any other tranche, except for the the lowest tranche. On the other hand,

for the exact method, calculating the spread for the j-th tranche takes as much time as

calculating the spreads for the first j tranches, provided that expected losses are calculated

for cumulative tranches, 0% to the various detachment points, and then differenced to obtain

results for individual tranches. This is the most efficient way to handle multiple tranches,

using the JKM method.

For a real deal with LGDs plotted in Figure 1, the exponential approximation method

runs faster than the recursive method JKM. Calculated with the JKM method, the spreads

for the first four tranches [0, 3], [3, 7], [7, 10] and [10, 15], are (2176.5, 643.3, 274.8, 122.4), with

16

Table 7: Ratio of CPU times used by the EAP method over those used by the JKM method,

to evaluate the fourth tranche, [10%, 15%], on the test pools

# terms in EAP

Pool ID 25 50 100 200 400

100-1 1.65 2.75 5.69 10.53 21.12

100-2 1.43 2.57 5.32 9.85 19.77

100-3 1.11 2.01 4.15 7.68 15.42

100-4 0.44 0.79 1.64 3.05 6.11

100-5 0.40 0.72 1.49 2.76 5.53

200-1 0.83 1.25 2.08 3.66 7.01

200-2 0.77 1.18 1.91 3.43 6.52

200-3 0.66 1.03 1.68 2.91 5.64

200-4 0.43 0.66 1.07 1.90 3.65

200-5 0.27 0.41 0.67 1.20 2.29

400-1 0.44 0.72 1.14 2.08 4.14

400-2 0.42 0.62 1.07 1.96 3.87

400-3 0.35 0.56 0.87 1.59 3.18

400-4 0.17 0.27 0.42 0.77 1.54

400-5 0.06 0.08 0.14 0.26 0.52

the calculation taking 7 seconds. Using the 100-term approximation method, we obtained the

same spreads, but used only 1.18 seconds; using the 50-term approximation we obtained the

spreads (2176.5, 643.2, 274.9, 122.5) in 0.65 second.

Comparisons of the accuracy of other methods, when applied to the main examples of

this section, are provided in [18]; see [22] for a description of the methods and other similar

examples.

17

0 20 40 60 80 100 120 1401

2

3

4

5

6

7

k

LGD

Figure 1: LGDs of the reference entities in a real portfolio, the values are relative to the

smallest LGD

3.5 Sensitivities

It is of practical interest to see how sensitive a tranche’s value or spread (when the latter is

not an input) is, to small changes in various parameters of the adopted model. Generally,

there are two approaches to sensitivity calculations: infinitesimal and macroscopic. In the

infinitesimal approach, one calculates the derivative of the value or spread with respect to

the parameter in closed form and then uses the derivative in a linear approximation. In

the macroscopic approach, the value or spread is recalculated at a slightly perturbed level

of the chosen parameter, and then the sensitivity is taken as the difference between the new

and original values or spreads. In principle, the infinitesimal approach can be carried out

for the EAP method, based on the formulae (15)-(18), just using the usual rules of calculus,

especially the chain rule. However, the resulting expression may be very complicated and

it is quite possible that the large number of arithmetic operations involved, will be slower

than a simple revaluation, especially if the invariant parts of the expression (18) are cached

18

and available during the revaluation. Of course, the identity of the parameter being tweaked,

has a strong influence on the result of the comparison of the two approaches. For example,

the formula resulting from tweaking a LGD is much less complicated than one resulting from

tweaking a PD. For the sake of generality, we will follow the macroscopic approach and restrict

attention to the accuracy of spread sensitivities calculated with the EAP method, for the sake

of concreteness, comparing the results to the benchmark JKM method.

Of all the model parameters that can be perturbed or tweaked, the probabilities of default

are the most interesting ones, especially as they affect all tranches.3 Specifically, we will

continue with the 15-pool examples of the previous section and study numerically the result

of subjecting the probablity of default (PD) curve to a positive parallel shift of 20 bp. The

results for the alternative choices of 10 bp and 5 bp, for the parallel shift, are similar and

described in detail in [19]. In Figures 2–5, the pool size 100 (respectively, 200, 400) is indicated

by a solid (respectively, dashed, dot-dashed) line while the pool type 1 (respectively, 2, 3, 4,

5) is indicated by the marker � (respectively, �, N, H, ◮).

We start with a 50-term EAP. Reported errors are relative and expressed as percentages,

in absolute value. The term base refers to prior to tweaking the PD curve while the term

tweaked refers to after tweaking the PD curve. The errors in the base values of the spreads of

the five tranches, for each of the 15 pools, are given in Figure 2. Next, the errors in the tweaked

values of the spreads of the five tranches, for each of the 15 pools, are given in Figure 3, along

with the errors in the sensitivities. The most striking feature is the reduction of the error in

the sensitivities, relative to the size of the errors in the base and tweaked values.

Since the largest errors are for the highest tranche, it is of interest to see how it behaves

as the number of terms in the EAP is increased. In Figures 4–5, the signed relative errors are

plotted for the base and tweaked spreads and the spread sensitivities.

3In practice, the default probabilities are themselves calculated quantities, being functions of hazard rates

which in turn are bootstrapped from market-quoted CDS spreads. For simplicity we will tweak the default

probabilities directly.

19

1 2 3 4 50

1

2

3

4

5

6

7

8

9

10

Relative Error in SpreadBase Value

tranche

% e

rror

100−1100−2100−3100−4100−5200−1200−2200−3200−4200−5400−1400−2400−3400−4400−5

Figure 2: Comparative results in percentage error for the 50-term EAP method for all five

tranches and 15 pools, K-m, where K = 100, 200, 400 and m = 1, 2, 3, 4, 5: base values.

4 Conclusion

A new method for pricing correlation-dependent derivatives has been proposed. The method

is based on an exponential approximation to the payoff function. With this approximation,

the evaluation of the conditional expectation of the stop-loss function of the credit portfolio

can be computed by calculating a series of conditional expectations for individual obligors. In

Section 3, we applied this method to synthetic CDO pricing where the correlation structure

of the underlying obligors is specified through a simple one-factor Gaussian copula model.

This method could be applied to other models belonging to the conditional independence

framework. From formula (8), we see that there are no restrictions on the distribution of each

Zk, other than the ease of calculation of its conditional Laplace transform.

The EAP method applies to the basic affine jump diffusion intensity-based models in-

troduced by Duffie and Garleanu [10] and extended succesively by Mortensen [27] and then

Eckner [11]. These models belong to the conditional independence framework with the auxil-

iary factor M (see Section 2 herein) being the integrated common component of the default

20

1 2 3 4 50

1

2

3

4

5

6

7

Relative Error in SpreadTweaked (20 bp) Value

tranche

% e

rror

100−1100−2100−3100−4100−5200−1200−2200−3200−4200−5400−1400−2400−3400−4400−5

1 2 3 4 50

0.5

1

1.5

2

2.5

Relative Error in Spread Sensitivity20−bp Tweak

tranche

% e

rror

100−1100−2100−3100−4100−5200−1200−2200−3200−4200−5400−1400−2400−3400−4400−5

Figure 3: Comparative results in percentage error for the 50-term EAP method for all five

tranches and 15 pools, K-m, where K = 100, 200, 400 and m = 1, 2, 3, 4, 5; tweak size: 20 bp.

21

50 100 200 400−10

−8

−6

−4

−2

0

2

Relative Error in SpreadBase Value

# EAP terms

% e

rror

100−1100−2100−3100−4100−5200−1200−2200−3200−4200−5400−1400−2400−3400−4400−5

Figure 4: Comparative results in percentage signed error for the 50-, 100-, 200-, and 400-

term EAP method for the Senior tranche and 15 pools, K-m, where K = 100, 200, 400 and

m = 1, 2, 3, 4, 5: base values.

intensities. There are explicit formulas available for the conditional default probabilities and

the probability density function of M can be obtained by inverting its Fourier transform,

which is known in a semi-analytic form – see the cited references for details. (In [10], there is

also a so-called sectoral extension of the model. In that case, M would be a random vector,

consisting of the independent, integrated sectoral default intensities as well as the single global

one. As such, a multidimensional numerical integration would be required in the last uncon-

ditioning step of either approach.) For the calibration of these models, it is efficient to use a

low-accuracy EAP (e.g., a 25-term one), in the first stage of the calibration. Once the solution

is close to the optimal one, one can use a high-accuracy EAP to speed up the convergence.

Note that the EAP approach avoids the necessity of rounding losses, which is a property of

the algorithm of [2] that is used in [27] and [11] to calculate the pool loss distribution.

The EAP approach also applies to more general models that can, e.g., incorporate stochas-

tic recovery rates as in [1] or [24]. In [1], the recovery rate is constant, conditional on the

auxilliary factor (latent credit driver), X. In [24], although LGD and 1{τ≤t} are correlated, for

22

50 100 200 400−7

−6

−5

−4

−3

−2

−1

0

1

Relative Error in SpreadTweaked (20 bp) Value

# EAP terms

% e

rror

100−1100−2100−3100−4100−5200−1200−2200−3200−4200−5400−1400−2400−3400−4400−5

50 100 200 400−0.5

0

0.5

1

1.5

2

2.5

Relative Error in Spread Sensitivity20−bp Tweak

# EAP terms

% e

rror

100−1100−2100−3100−4100−5200−1200−2200−3200−4200−5400−1400−2400−3400−4400−5

Figure 5: Comparative results in percentage signed error for the 50-, 100-, 200-, and 400-

term EAP method for Senior tranche and 15 pools, K-m, where K = 100, 200, 400 and

m = 1, 2, 3, 4, 5; tweak size: 20 bp.

23

each obligor, the LGD and the indicator function depend on the same Y , the creditworthiness

of the obligor; and the definition of LGD is explicit. Thus we can compute the conditional

expectation, E[LGD1{τ≤t} |X], explicitly.

Here is a summary of the advantages and disadvantages of the EAP method, as applied

to synthetic CDOs.

Advantages of EAP

• It is faster than the traditional (loss distribution) approach in any (hence, especially

simultaneously all) of the following situations: single tranches; very heterogeneous pools;

large pools. The CPU time for the EAP method is independent of the pool structure:

its computational cost depends only on the number of names in the pool and the number

of terms in the exponential approximation.

• It is quite accurate; e.g., with 50 exponential terms, spreads are observed to be correct

to within 1 bp, representing a 0.5% error for all but the highest tranches where the

errors are of the same order of magnitude as the spreads.

• It does not round losses, as in many versions of the traditional approach which round

losses to values on a regular grid (e.g., for use with FFT).

• The weights and exponents can be calculated once, stored, then used for many pools

and tranches. Compared to the saddlepoint approximation method, this is the main

advantage of EAP, as the saddlepoint method must compute some parameters dynami-

cally.

• In the EAP method, the number of terms used can be tailored to each tranche with a

higher number of terms used for higher tranches, to improve accuracy.

• It easily handles numerical sensitivities (e.g., to PDs) and is quite accurate, even for the

highest tranches.

24

Disadvantages (Scope) of EAP

• It is slower than the traditional approach in any of the following situations: evaluation

of multiple (> 3) tranches on a single pool; the highest tranches (requiring around 200

terms for comparable accuracy, ceteris paribus [such as no rounding of losses]); very

homogeneous pools.

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